Question:
Evaluate
\[
\int_0^\pi \left( \sin^3 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^3 x \right) dx = ?
\]
Evaluate
\[
\int_0^\pi \left( \sin^3 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^3 x \right) dx = ?
\]
Show Hint
Simplify the integrand using trigonometric identities and then apply integration techniques to solve.
Updated On: July 22, 2025
- \(\frac{873}{2240}\)
- \(\frac{3\pi}{12}\)
- \(\frac{1641}{4480}\)
- \(\frac{3\pi}{128}\)
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The Correct Option is A
Solution and Explanation
Use trigonometric identities to simplify the integrand:
\[
\sin^3 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^3 x = 2\sin^3 x \cos^3 x + \sin^4 x \cos^4 x.
\]
This integral can be split into two parts, and standard methods of integration yield the result:
\[
\frac{873}{2240}.
\]
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